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Theorem mvrsval 30656
 Description: The set of variables in an expression. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
mvrsval.v 𝑉 = (mVR‘𝑇)
mvrsval.e 𝐸 = (mEx‘𝑇)
mvrsval.w 𝑊 = (mVars‘𝑇)
Assertion
Ref Expression
mvrsval (𝑋𝐸 → (𝑊𝑋) = (ran (2nd𝑋) ∩ 𝑉))

Proof of Theorem mvrsval
Dummy variables 𝑡 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mvrsval.w . . 3 𝑊 = (mVars‘𝑇)
2 elfvex 6131 . . . . 5 (𝑋 ∈ (mEx‘𝑇) → 𝑇 ∈ V)
3 mvrsval.e . . . . 5 𝐸 = (mEx‘𝑇)
42, 3eleq2s 2706 . . . 4 (𝑋𝐸𝑇 ∈ V)
5 fveq2 6103 . . . . . . 7 (𝑡 = 𝑇 → (mEx‘𝑡) = (mEx‘𝑇))
65, 3syl6eqr 2662 . . . . . 6 (𝑡 = 𝑇 → (mEx‘𝑡) = 𝐸)
7 fveq2 6103 . . . . . . . 8 (𝑡 = 𝑇 → (mVR‘𝑡) = (mVR‘𝑇))
8 mvrsval.v . . . . . . . 8 𝑉 = (mVR‘𝑇)
97, 8syl6eqr 2662 . . . . . . 7 (𝑡 = 𝑇 → (mVR‘𝑡) = 𝑉)
109ineq2d 3776 . . . . . 6 (𝑡 = 𝑇 → (ran (2nd𝑒) ∩ (mVR‘𝑡)) = (ran (2nd𝑒) ∩ 𝑉))
116, 10mpteq12dv 4663 . . . . 5 (𝑡 = 𝑇 → (𝑒 ∈ (mEx‘𝑡) ↦ (ran (2nd𝑒) ∩ (mVR‘𝑡))) = (𝑒𝐸 ↦ (ran (2nd𝑒) ∩ 𝑉)))
12 df-mvrs 30640 . . . . 5 mVars = (𝑡 ∈ V ↦ (𝑒 ∈ (mEx‘𝑡) ↦ (ran (2nd𝑒) ∩ (mVR‘𝑡))))
13 fvex 6113 . . . . . . 7 (mEx‘𝑇) ∈ V
143, 13eqeltri 2684 . . . . . 6 𝐸 ∈ V
1514mptex 6390 . . . . 5 (𝑒𝐸 ↦ (ran (2nd𝑒) ∩ 𝑉)) ∈ V
1611, 12, 15fvmpt 6191 . . . 4 (𝑇 ∈ V → (mVars‘𝑇) = (𝑒𝐸 ↦ (ran (2nd𝑒) ∩ 𝑉)))
174, 16syl 17 . . 3 (𝑋𝐸 → (mVars‘𝑇) = (𝑒𝐸 ↦ (ran (2nd𝑒) ∩ 𝑉)))
181, 17syl5eq 2656 . 2 (𝑋𝐸𝑊 = (𝑒𝐸 ↦ (ran (2nd𝑒) ∩ 𝑉)))
19 fveq2 6103 . . . . 5 (𝑒 = 𝑋 → (2nd𝑒) = (2nd𝑋))
2019rneqd 5274 . . . 4 (𝑒 = 𝑋 → ran (2nd𝑒) = ran (2nd𝑋))
2120ineq1d 3775 . . 3 (𝑒 = 𝑋 → (ran (2nd𝑒) ∩ 𝑉) = (ran (2nd𝑋) ∩ 𝑉))
2221adantl 481 . 2 ((𝑋𝐸𝑒 = 𝑋) → (ran (2nd𝑒) ∩ 𝑉) = (ran (2nd𝑋) ∩ 𝑉))
23 id 22 . 2 (𝑋𝐸𝑋𝐸)
24 fvex 6113 . . . . 5 (2nd𝑋) ∈ V
2524rnex 6992 . . . 4 ran (2nd𝑋) ∈ V
2625inex1 4727 . . 3 (ran (2nd𝑋) ∩ 𝑉) ∈ V
2726a1i 11 . 2 (𝑋𝐸 → (ran (2nd𝑋) ∩ 𝑉) ∈ V)
2818, 22, 23, 27fvmptd 6197 1 (𝑋𝐸 → (𝑊𝑋) = (ran (2nd𝑋) ∩ 𝑉))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∩ cin 3539   ↦ cmpt 4643  ran crn 5039  ‘cfv 5804  2nd c2nd 7058  mVRcmvar 30612  mExcmex 30618  mVarscmvrs 30620 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-mvrs 30640 This theorem is referenced by:  mvrsfpw  30657  msubvrs  30711
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